Question Number 190448 by Abdullahrussell last updated on 03/Apr/23
Answered by a.lgnaoui last updated on 04/Apr/23
![a+b+c =6 (1) ab+bc+ca=3 (2) abc =−1 (3) (1)×(2)⇒ a^2 b+abc+a^2 c+ab^2 + +cb^2 +abc+abc+bc^2 +ac^2 =3(abc)+(a^2 b+b^2 c+c^2 a)+ (ab^2 +bc^2 +ca^2 ) 21=(a^2 b+b^2 c+c^2 a)+(ab^2 +bc^2 +ca^2 )(4) (a+b+c)^2 =(a^2 +b^2 +c^2 )+6 a^2 +b^2 +c^2 =30 (5) a^3 +b^3 +c^3 =(a+b+c)^3 − 3[(a+b)+c][c(a+b)] =6^3 −3(6)(ac+ab) a^3 +b^3 +c^3 =216−18bc (6) (a+b+c)(a^2 +b^2 +c^2 ) =a^3 +b^3 +c^3 +ab^2 +ac^2 + ba^2 +bc^2 +ca^2 +cb^2 =216−18bc+21=237−18bc =6×30=180 18bc=237−180 ⇒ bc=((19)/6) abc=−1⇒ a=((−6)/(19)) { ((a=((−6)/(19)) bc=((19)/6))),((a+(b+c)=6)) :} b+c=6+(6/(19))=((120)/(19)) { ((b+c=((120)/(19)))),((bc =((19)/6))) :} z^2 −((120)/(19))z+((19)/6)=0 △=(((60)/(19)))^2 −((19)/6) =((3600×6−19^3 )/(19^2 ×6))= z=(0,55 ; 5,766) (a,b ,c)={(((−6)/(19)) , 0,55, 5,766);(((−6)/(19)),5,766, 0,55)} calcul de: a^2 b+b^2 c+c^2 a ((36)/(19^2 ))×0,55+0,55^2 ×5,766−5,766^2 ×(6/(19)) a^2 b+b^2 c+c^2 a=−8,65](https://www.tinkutara.com/question/Q190497.png)
$$\:{a}+{b}+{c}\:\:\:\:\:\:=\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\:{ab}+{bc}+{ca}=\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$$\:{abc}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\mathrm{1}\:\:\:\:\:\:\:\left(\mathrm{3}\right) \\ $$$$ \\ $$$$\left(\mathrm{1}\right)×\left(\mathrm{2}\right)\Rightarrow\:{a}^{\mathrm{2}} {b}+{abc}+{a}^{\mathrm{2}} {c}+{ab}^{\mathrm{2}} + \\ $$$$\:\:\:\:\:+{cb}^{\mathrm{2}} +{abc}+{abc}+{bc}^{\mathrm{2}} +{ac}^{\mathrm{2}} \\ $$$$\:\:\:=\mathrm{3}\left({abc}\right)+\left({a}^{\mathrm{2}} {b}+{b}^{\mathrm{2}} {c}+{c}^{\mathrm{2}} {a}\right)+ \\ $$$$\:\:\:\:\:\:\left({ab}^{\mathrm{2}} +{bc}^{\mathrm{2}} +{ca}^{\mathrm{2}} \right) \\ $$$$\mathrm{21}=\left({a}^{\mathrm{2}} {b}+{b}^{\mathrm{2}} {c}+{c}^{\mathrm{2}} {a}\right)+\left({ab}^{\mathrm{2}} +{bc}^{\mathrm{2}} +{ca}^{\mathrm{2}} \right)\left(\mathrm{4}\right) \\ $$$$ \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} =\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)+\mathrm{6} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{30}\:\:\:\:\:\left(\mathrm{5}\right) \\ $$$$ \\ $$$$\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\left({a}+{b}+{c}\right)^{\mathrm{3}} − \\ $$$$\mathrm{3}\left[\left({a}+{b}\right)+{c}\right]\left[{c}\left({a}+{b}\right)\right] \\ $$$$=\mathrm{6}^{\mathrm{3}} −\mathrm{3}\left(\mathrm{6}\right)\left({ac}+{ab}\right) \\ $$$$ \\ $$$$\:\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{216}−\mathrm{18}{bc}\:\:\left(\mathrm{6}\right) \\ $$$$ \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right) \\ $$$$={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{ab}^{\mathrm{2}} +{ac}^{\mathrm{2}} + \\ $$$${ba}^{\mathrm{2}} +{bc}^{\mathrm{2}} +{ca}^{\mathrm{2}} +{cb}^{\mathrm{2}} \\ $$$$=\mathrm{216}−\mathrm{18}{bc}+\mathrm{21}=\mathrm{237}−\mathrm{18}{bc} \\ $$$$=\mathrm{6}×\mathrm{30}=\mathrm{180} \\ $$$$\:\:\mathrm{18}{bc}=\mathrm{237}−\mathrm{180}\:\:\Rightarrow\:{bc}=\frac{\mathrm{19}}{\mathrm{6}} \\ $$$${abc}=−\mathrm{1}\Rightarrow\:\:\:\:{a}=\frac{−\mathrm{6}}{\mathrm{19}} \\ $$$$ \\ $$$$\begin{cases}{{a}=\frac{−\mathrm{6}}{\mathrm{19}}\:\:\:\:{bc}=\frac{\mathrm{19}}{\mathrm{6}}}\\{{a}+\left({b}+{c}\right)=\mathrm{6}}\end{cases} \\ $$$${b}+{c}=\mathrm{6}+\frac{\mathrm{6}}{\mathrm{19}}=\frac{\mathrm{120}}{\mathrm{19}} \\ $$$$\:\:\:\begin{cases}{{b}+{c}=\frac{\mathrm{120}}{\mathrm{19}}}\\{{bc}\:\:\:\:=\frac{\mathrm{19}}{\mathrm{6}}}\end{cases} \\ $$$${z}^{\mathrm{2}} −\frac{\mathrm{120}}{\mathrm{19}}{z}+\frac{\mathrm{19}}{\mathrm{6}}=\mathrm{0}\:\:\:\bigtriangleup=\left(\frac{\mathrm{60}}{\mathrm{19}}\right)^{\mathrm{2}} −\frac{\mathrm{19}}{\mathrm{6}} \\ $$$$=\frac{\mathrm{3600}×\mathrm{6}−\mathrm{19}^{\mathrm{3}} }{\mathrm{19}^{\mathrm{2}} ×\mathrm{6}}= \\ $$$$\:\:\:{z}=\left(\mathrm{0},\mathrm{55}\:;\:\:\:\mathrm{5},\mathrm{766}\right)\:\:\: \\ $$$$\left({a},{b}\:,{c}\right)=\left\{\left(\frac{−\mathrm{6}}{\mathrm{19}}\:\:,\:\:\:\mathrm{0},\mathrm{55},\:\:\:\:\mathrm{5},\mathrm{766}\right);\left(\frac{−\mathrm{6}}{\mathrm{19}},\mathrm{5},\mathrm{766},\:\:\:\mathrm{0},\mathrm{55}\right)\right\} \\ $$$$\boldsymbol{{calcul}}\:\boldsymbol{{de}}:\:{a}^{\mathrm{2}} {b}+{b}^{\mathrm{2}} {c}+{c}^{\mathrm{2}} {a} \\ $$$$\:\frac{\mathrm{36}}{\mathrm{19}^{\mathrm{2}} }×\mathrm{0},\mathrm{55}+\mathrm{0},\mathrm{55}^{\mathrm{2}} ×\mathrm{5},\mathrm{766}−\mathrm{5},\mathrm{766}^{\mathrm{2}} ×\frac{\mathrm{6}}{\mathrm{19}} \\ $$$$\boldsymbol{{a}}^{\mathrm{2}} \boldsymbol{{b}}+\boldsymbol{{b}}^{\mathrm{2}} \boldsymbol{{c}}+\boldsymbol{{c}}^{\mathrm{2}} \boldsymbol{{a}}=−\mathrm{8},\mathrm{65} \\ $$